cp's OEIS Frontend

A332305 Number of compositions (ordered partitions) of n into distinct parts such that number of parts is even.

%I A332305 #8 Feb 13 2020 11:50:49
%S A332305 1,0,0,2,2,4,4,6,6,8,32,34,58,84,132,158,230,280,376,450,570,1388,
%T A332305 1556,2398,3310,4920,6600,9674,12122,16684,21340,28110,34974,45392,
%U A332305 55208,69274,124498,143676,204012,270758,377966,493024,690304,895434,1223826,1562948
%N A332305 Number of compositions (ordered partitions) of n into distinct parts such that number of parts is even.
%H A332305 Alois P. Heinz, <a href="/A332305/b332305.txt">Table of n, a(n) for n = 0..5000</a>
%H A332305 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A332305 G.f.: Sum_{k>=0} (2*k)! * x^(k*(2*k + 1)) / Product_{j=1..2*k} (1 - x^j).
%F A332305 a(n) = A032020(n) - A332304(n).
%e A332305 a(5) = 4 because we have [4, 1], [3, 2], [2, 3] and [1, 4].
%p A332305 b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0,
%p A332305       irem(p+1, 2)*p!, add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
%p A332305     end:
%p A332305 a:= n-> b(n$2, 0):
%p A332305 seq(a(n), n=0..55);  # _Alois P. Heinz_, Feb 09 2020
%t A332305 nmax = 45; CoefficientList[Series[Sum[(2 k)! x^(k (2 k + 1))/Product[1 - x^j, {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]
%Y A332305 Cf. A027187, A032020, A034008, A067661, A332304.
%K A332305 nonn
%O A332305 0,4
%A A332305 _Ilya Gutkovskiy_, Feb 09 2020